In this type of fanfiction, Snape could simply intimidate Vernon. Snape has an air of mystery and a certain power about him. He might show up at Vernon's doorstep, give Vernon a cold stare, and with his reputation as a powerful wizard, Vernon might be too scared to oppose him. Snape could then just take Harry without much resistance from Vernon.
In this fan fiction scenario, Snape might use his knowledge of the Muggle world and Vernon's weaknesses. Snape could have found out some information about Vernon's business affairs or personal secrets. Then he blackmails Vernon into leaving Harry alone. By doing this, he effectively saves Harry from Vernon without having to use too much magic and draw unwanted attention.
Perhaps Dudley finally stands up to his father's cruel treatment of Harry. Maybe he realizes that Vernon's behavior towards Harry is wrong and uses his own physical strength or some clever ruse to get Harry out of a difficult situation Vernon has put him in.
The Dursleys might be left in a state of shock and confusion. Since they have been so mean to Harry, they could be worried that the wizarding world will come after them for revenge. But since they are Muggles, the wizarding world may just leave them be, but they will always live in fear of the unknown.
In the canon story, Snape has a very complex and often antagonistic relationship with Harry. But in fanfictions where he saves baby Harry, it's a huge departure. Here, Snape starts off as a protector, which is completely different from his role in the books where he seems to begrudgingly help Harry at times. It also changes the power dynamics as Harry may grow up with a sense of debt or gratitude towards Snape, rather than the suspicion and dislike he had in the original.
The idea of Severus Snape saving baby Harry Potter in fanfiction is really fascinating. It gives authors the chance to explore a lot of uncharted territory. Snape, who was so conflicted in the original books, can be shown as more of a hero in this scenario. It can also change the whole Hogwarts experience for Harry. Instead of the initial hostility between them, there could be a sense of respect or even a mentorship. And it might even affect the relationships Harry has with other characters, like the way he views the Slytherin house.
In fanfiction, Snape saving Harry often showcases a complex side of Snape's character. It might be a redemption arc for him, as in the original series, Snape had a very ambiguous relationship with Harry. In these fanfictions, it could be due to his love for Lily that he finally decides to fully protect Harry, which is a very interesting take on his character.
One motivation could be Snape's sense of duty as a teacher. Despite his dislike for Harry in some ways, he still has an obligation to protect the students. So, when he sees Lockhart endangering Harry, he steps in.